---
title: "Energy/power signals; periodicity of cos + sin"
subject: "Digital Signal Analysis and Processing"
chapter: 1
key: "semester_7/digital_signal_analysis_and_processing:energy-power-periodic-2"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/energy-power-periodic-2
site: BCT Engineering
---

# Energy/power signals; periodicity of cos + sin

Digital Signal Analysis and Processing · Chapter 1: Discrete Time Signals and Systems

- Asked in: 2069 Chaitra, 2+2 marks
- Video: [Determine signal is    periodic or aperiodic  (Digital signal processing)](https://www.youtube.com/watch?v=wpLphB7vZEo) — Last moment tuitions
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/energy-power-periodic-2
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-1

## Question

Define energy and power type signal with suitable example. Check whether the signal x[n] = cos(2πn/5) + sin(πn/3) is periodic or not.

## Short answer

Energy signal: finite energy, zero power. Power signal: finite nonzero power, infinite energy. x[n]=cos(2πn/5)+sin(πn/3) is periodic because both components have rational frequency ratios (1/5 and 1/6). Fundamental period is LCM(5,6)=30 samples.

## Answer

#### Energy and power signals
- **Energy signal:** finite energy E = Σ |x[n]|² (n from −∞ to ∞), 0 < E < ∞, and average power P = 0. Example: x[n] = (0.5)ⁿ u[n] has E = 1/(1 − 0.25) = 4/3.
- **Power signal:** finite non-zero average power P = lim(N→∞) 1/(2N+1) Σ(n=−N to N) |x[n]|², with infinite energy. Example: x[n] = cos(ω₀n) has P = ½; the unit step has P = ½.

#### Periodicity of x[n] = cos(2πn/5) + sin(πn/3)
A sum of periodic sequences is periodic if the individual periods have a common multiple. Check each term (each ω₀/2π must be a ratio of integers):

- cos(2πn/5): ω₁ = 2π/5, f₁ = 1/5, so N₁ = 5.
- sin(πn/3): ω₂ = π/3, f₂ = 1/6, so N₂ = 6.

Both frequencies are rational, so the sum is **periodic**. The fundamental period is the least common multiple:

N = LCM(5, 6) = **30 samples**.

Check: x[n + 30] contains cos(2πn/5 + 12π) and sin(πn/3 + 10π), both equal to the original values.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A power signal has
   A. finite energy and zero power
   B. infinite energy and finite non-zero average power
   C. infinite energy and infinite power
   D. zero energy
   Answer: B. Power signals have E = infinity but P finite and non-zero, e.g. the unit step with P = 1/2. (Key point: Power signal definition)

2. Which signal is an energy signal?
   A. x[n] = (0.5)^n u[n]
   B. x[n] = u[n]
   C. x[n] = cos(w0 n)
   D. x[n] = n
   Answer: A. (0.5)^n u[n] has E = 1/(1 - 0.25) = 4/3, which is finite. (Key point: Energy signal example)

3. For cos(2 pi n/5) and sin(pi n/3), the individual periods are
   A. 5 and 3
   B. 2.5 and 6
   C. 10 and 6
   D. 5 and 6
   Answer: D. cos: f = 1/5 gives N = 5; sin: f = 1/6 gives N = 6. (Key point: Period of each term)

4. The fundamental period of the sum is
   A. 5 + 6 = 11
   B. The smaller one, 5
   C. LCM(5, 6) = 30
   D. 5 x 6 x 2 = 60
   Answer: C. The sum repeats at the least common multiple of the individual periods, 30 samples. (Key point: Sum of periodic signals)

## Related in Digital Signal Analysis and Processing

- [Energy/power signals; periodicity of complex exponential](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aenergy-power-periodic-1) — Digital Signal Analysis and Processing, chapter 1
- [Periodic signals and fundamental period](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aperiodic-signals-fundamental-period) — Digital Signal Analysis and Processing, chapter 1
- [Energy or power: u[n] and δ[n]](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aenergy-power-unit-step-impulse) — Digital Signal Analysis and Processing, chapter 1

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